Optimization Definition and 588 Threads

Homework Statement I remember doing something very similar to this in pre-calc, but I don't know where to get started. A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size will be cit out of each corner, and then the ends and sides will...

Homework Statement Minimize 2x²+2y²-2xy-9y subject 4x + 3y =,< 10 , y - 4x² =,< -2 x >,= 0 and y >,= 0. I don't undersant this: "This equation has no nonnegative root, which contradicts a nonnegativity constraint." and how we solve -16x² + 2x + 17 + h2 = 0

I'm having a tough time solving this question.:frown: I'd appreciate it if someone can please help me out Homework Statement Homework Equations y=x^2 Triangle QPR =3/4 of the area of the parabolic segment enclosed between QR and the parabola The Attempt at a Solution I...

How would you go about finding the highest point on the curve of intersection of an ellipsoid and a plane? Given: x^2 + y^2 + z^2 = k and ax + by + cz = j. I was thinking about using Lagrange Multipliers but I would always get stuck.

For a calc project, I am supposed to solve an interesting calculus word problem dealing with maximum and minimum values. The catch is that I cannot use my own book. Can anyone suggest a challenging optimization problem? So far, we've covered the problem with a person who must find the least time...

Homework Statement Each edge of a square has length L. Prove that among all squares inscribed in the given square, the one of minimum area has edges of length \frac{1}{2}L\sqrt{2} Homework Equations The Attempt at a Solution I started by drawing a square of sides L. Then labeled...

Homework Statement We are given a graph of gallons of fuel per hour versus miles per hour and asked what speed should be used to maximize fuel efficiency and also what is the optimal speed(are these two the same thing). Homework Equations I understand optimization using the first...

Homework Statement Homework Equations a^2 + b^2 = c^2 A= (1/2)bh The Attempt at a Solution 1)I labeled the distance traveled on the ocean as y and the distance traveled on land as x. 2)This one's kinda hard to describe: you know how the 100 yd distance and the shoreline form...

Homework Statement You're building a walkway from the corner of one building to the corner of another building. The diagram looks like this. The street is 100 ft wide, and 50 ft long. The walkway will weigh 40 pounds per feet when it is parallel to the street and 30 pounds per feet when it is...

Here is what what written about this Lean-To: A lean-to has a wooden floor, a 10 feet high open front, wooden side and back walls, and a wooden roof that tilts down at an angle of 45 degrees. What are the dimensions of the lean-to with the largest possible floor area that can be constructed...

Homework Statement Find the area of the largest rectangle that can be inscribed in the region bounded by the graph of y = (4-x)/(2+x) and the coordinate aces in the first quadrant. I think my only problem with this one is taking the derivative, this is what i get y' = (-x^2 - 4x +...

Homework Statement x^2 - 2xy + 6y^2 = 10 Find the point on the ellipse closest to the origin (0,0).Homework Equations The Attempt at a Solution Absolutely no one in my class can solve this. We've been to the math lab and none of the helpers there know how to solve it. I think the only person...

Homework Statement At which points on the curve y=1 + 40x^3 - 3x^5 does the tangent line have th largest slope? Homework Equations derivative is 120x^2-15x^4... The Attempt at a Solution how should i do this? start by setting the derivative equal to zero to find critical #s, but...

Homework Statement This is from the book Calculus Single Variable, 4th edition: The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius <i>r</i>. If the cross-sectional area is A, determine the dimensions of the cross section which...

I'm stuck with a problem of finding the minimum energy of an inorganic crystal. Crystal is of monoclinic structure with 18 parameters to specify its internal coordinates and 2 atomic species. Am using a classical interatomic potential found in literature. The minimization involves both the...

Homework Statement According to postal regulations, a carton is classified as "over-sized" if the sum of its height and girth (the perimeter of its base) exceeds 108 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume. Homework Equations...

Homework Statement Find the angle theta that maximizes the area of an isosceles triangle whose legs have length l. The angle is the top angle if the left and right sides are l coming to a point with the bottom leg horizontal. Homework Equations The Attempt at a Solution I broke...

:cry:Sorry to ask such a question, but our study group is at a loss as how to continue and our homework is due tomorrow. So here goes: In order to receive credit we MUST use calculus techniques: We have a piece of wire that is 100cm long and we're going to cut it into two pieces. One piece...

This might be a stupid question, I'm not use to asking questions about math... I just started on optimization. Can someone tell me What optimization is used for and how I could apply it to a problem. When it comes to a problem, I could do it like if it asks " Find two numbers that satisfy the...

Find the maximum volume of a right circular cone placed upside down in a right circular cone of radius R and height H. The volume of a cone of radius r and height h is 4/3pir^2h. I need help starting this one up... Any feedback will be appreciated.

Find the point in the plane 3x+2y+z=1 that is the closest to the origin by minimising squared distance. (I hope I translated this ok..) I was thinking I would need to isolate a variable in the equation for the plane above then substitute it into the distance formula then do a partial...

Homework Statement The Question is: "A 400km racetrack is to be built with two straight sides and semicricles at the ends. Find the dimensions of the track that encloses the maximum area." The two long sides of the rectangle are written with >/= to 100m (each) The straight side of the 2 semi...

A dune buggy is on a straight desert road, 40 km north of Dust city. The vehicle can travel at 45 km/h off the road and 75 km/h on the road. The driver wants to go to Gulch city, 50 km east of Dust city in the shortest possible time. Determine the route he should take. The equation i can up...

Homework Statement A right circular cylinder in inscribed in a cone with height 10 and base radius 3. Find the largest possible voluem of such a cylinder. Homework Equations V=\pi*r^2*h The Attempt at a Solution Ok, so I used similar triangles of the cone and cylinder to obtain...

Homework Statement A cylindrical can is to hold 500 cm^3 of apple juice the design must take into account that the height must be between 6 and 15 cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? assume no waste...

Optimization, Minima, new question:Sheet Alluminum Homework Statement A box with a square base and no top must haave a volume of 10000 cm^3. If the smallest dimension in any direction is 5 cm, then determine the dimensions of the box that minimize the amount of material used. Homework...

Homework Statement A man with a boat is located at point P on the shore of a circular lake of radius 5 miles. He wants to reach the point Q on the shore diametrically opposed to P as quickly as possible. He plans to paddle his boat at an angle t(0<t<pi/2)<or equal to** to PQ to some point...

This problem has to do with physics but it is from my calculus book, and for my calc class so I put it here: Homework Statement "A component is designed to slide a block of steel with weight W across a table and into a chute. The motion of the block is resisted by a frictional force...

Hi there I was just wondering how to type Optimatiozion of a Parametric equation, in LaTeX?

I have a kite pictured below...i need to determine x y z so that the kite has a maximum area. a and b are fixed constants. The way I am thinking of trying this is to have this equation (x*y)+(z*x)=A from this i have three variables so i should solve for y in terms of x and solve for z in...

I have this optimization problem, with the solution, but I don't really understand how to do this. Can someone please explain it to me? I mean, I the solution, I got totally lost when he started working out the problem after that long paragraph. Where did he get the first equation from? One...

I was doing some optimization questions to get ready for a test. I came across one that stumped me. The question was "Find the dimensions of the isosceles triangle of the largest area that can be inscribed in a circle of radius r". My approach was: let y be the base of triangle let x be...

ok, i have an aquarium of volume V... where the cost of the base is 5 times of the sides... let h- height , w- width, l - lenght... therefore 2hw + 2hl is the cost of the sides and 5wl is the cost of the base therefore total cost = 2hw + 2hl + 5wl --> min and i want find the dimensions to...

Well Strange to me at least, A man wants to build a patio on his house in the shape of an isosceles triangle. He wants to build the side walls out of pink planks, but he has only 600 yards worth of planks. Find the dimensions of the largest area he can build if he's using the side of his...

two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a Point R on the ground to the top of the second polle. Show that the shortest length of rope occurs when \theta_1=\theta_2 I have found eqn and found the derivative, what do I do to show that it...

V = 2/3 pi r³ h 200 = 2/3 pi r³ h 300 = pi r³ h h = 300/ (pi)(r³) So this is the relationship that I find between height and radius. This is where I'm lost. The pricing of the cynlindrical wall and the hemisphere :S If I can get the formula for this one, I think I can solve the problem/...

Given 80 feet of fencing, what is the maximum area that you can enclose along a wall? Solution: L=lenght, W=width, A=area 2L+2W=80 (perimeter) ==> L+W=40 ==> L=40-W LW=A ==> (40-W)(W)= -W^2+ 40W= A (dA/dW)=0=-2W+40=0 ==> W=20 W=20...

A 51 meter length of wire is cut into two parts. The first part is fashioned into a rectangle that is twice as long as it is wide. The second part is fashioned into a square. How much of the originial wire is used for each shape if the shapes' combined area is a minimum? Use the...

Any recommendations? The books I have are very outdated. Extremely important to me are: - worked examples #1 criteria. Need that bridge between theory and implementation. - not overly heavy on theory (don't want to hire a PhD to explain it). I have an MS Engineering level education...

http://img117.imageshack.us/img117/3055/diagram28do.jpg I drew out that diagram but I think I might be wrong. The question is: Tom makes an open box from a rectangular piece of metal by cutting equal squares from the four corners and turning up the sides. The piece of metal measures 60 x...

We're doing optimization problems and I was just wondering if I set this one up right: What are the dimensions of a rectangle with an area of 64m^2 and the smallest possible perimeter? Area=xy 64=xy y=64/x Perimeter=2x+2y = 2x+2(64/x) = 2x+128/x

The Illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If the two light sources, one three times as strong as the other, are placed 10ft apart, where should an object be...

I need to "write the number 120 as a sum of three numbers so that the sum of the products taken two at a time is a maximum." I think this means that x+y+z=120 and xy+xz+yz=maximum. Can someone help me begin this problem?

Ok, well the problem states: A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?The first step that I took was to draw a picture. I just drew a semicircle with a...

Im not sure if this is the right place, but I have an optimization problem where I assume we are supposed to use the Lagraingian method: Consider the labour supply problem for an individual over an entire year. Suppose the individuals utility is described by the function U = (C^0.5) x...

Hi everyone, Well its that great time of year...the time of feverishly studying for finals and I have been doing some practice questions and there are a few that I'm stuck on. My first question I am embarassed to ask, it should be so simple, yet I cannot get the right answer. 1) A highway...

A particle is traveling along the postivie x-axis at a constant speed of 5 units per second. a) Where is the point when its distance from the point (0, 1) is increasing at a rate of 4 units per second? b) Where is the point when its distance from the point (0, 1) is increasing at a rate of...

Hi all, I'm having trouble with the following problem: It was given as a word problem from which to infer the mathematics but basically it is this: Maximize: f(x,y,z,t,w)= ln((y^2-x^2)(z^2-t^2)*w^3)+.8x-1.2y-20z/17+14t/17-w^3/(pi^3) Subject to the constraints: 0<= .5x+y+3z+3y+2.5w<+30...

Hello Everyone, I'm working on optimization problems right now, and let me tell ya, if you thought I was horrible at everything else wait until you see me attempt one of these I have three questions but I'm hoping if I just get help with the one I will be able to get the processes for the...

can someone PLEASE help me wit this problem? i will be ETERNALLY GRATEFUL. THANK YOU!. A light is suspended at a height "h" above the floor. The illumination at the point P is inversely proportional to the square of the distance from the point P to the light ("r") and directly proportional...